m-system

Let R be a ring. A subset S of R is called an m-system if

•

S≠∅, and

•

for every two elementsx,y∈S, there is an element r∈R such that x⁢r⁢y∈S.

m-Systems are a generalization of multiplicatively closet subsets in a ring. Indeed, every multiplicatively closed subset of R is an m-system: any x,y∈S, then x⁢y∈S, hence x⁢y⁢y∈S. However, the converse is not true. For example, the set

{rn∣r∈R⁢ and ⁢n⁢ is an odd positive integer}

is an m-system, but not multiplicatively closed in general (unless, for example, if r=1).

Remarks. m-Systems and prime ideals of a ring are intimately related. Two basic relationships between the two notions are

1.

An ideal P in a ring R is a prime ideal iff R-P is an m-system.

Proof.

P is prime iff x⁢R⁢y⊆P implies x or y∈P, iff x,y∈R-P implies that there is r∈R with x⁢r⁢y∉P iff R-P is an m-system.
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2.

Given an m-system S of R and an ideal I with I∩S=∅. Then there exists a prime ideal P⊆R with the property that P contains I and P∩S=∅, and P is the largest among all ideals with this property.

Proof.

Let 𝒞 be the collection of all ideals containing I and disjoint from S. First, I∈𝒞. Second, any chain K of ideals in 𝒞, its union ⋃K is also in 𝒞. So Zorn’s lemma applies. Let P be a maximal element in 𝒞. We want to show that P is prime. Suppose otherwise. In other words, a⁢R⁢b⊆P with a,b∉P. Then ⟨P,a⟩ and ⟨P,b⟩ both have non-empty intersections with S. Let

c=p+f⁢a⁢g∈⟨P,a⟩∩S and d=q+h⁢b⁢k∈⟨P,b⟩∩S,

where p,q∈P and f,g,h,k∈R. Then there is r∈R such that c⁢r⁢d∈S. But this implies that